Collocation methods for differential equations with piecewise linear delays

Hui Liang; Hermann Brunner

doi:10.3934/cpaa.2012.11.1839

Collocation methods for differential equations with piecewise linear delays

Hui Liang^*, Hermann Brunner

^*Corresponding author for this work

Department of Mathematics

Research output: Contribution to journal › Journal article › peer-review

8 Citations (Scopus)

Abstract

After analyzing the regularity of solutions to delay differential equations (DDEs) with piecewise continuous (linear) non-vanishing delays, we describe collocation schemes using continuous piecewise polynomials for their numerical solution. We show that for carefully designed meshes these collocation solutions exhibit optimal orders of global and local superconvergence analogous to the ones for DDEs with constant delays. Numerical experiments illustrate the theoretical superconvergence results.

Original language	English
Pages (from-to)	1839-1857
Number of pages	19
Journal	Communications on Pure and Applied Analysis
Volume	11
Issue number	5
DOIs	https://doi.org/10.3934/cpaa.2012.11.1839
Publication status	Published - Sept 2012

Scopus Subject Areas

Analysis
Applied Mathematics

User-Defined Keywords

Collocation methods
Delay differential equations
Optimal order of superconvergence
Piecewise non-vanishing delays
Regularity of solutions

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10.3934/cpaa.2012.11.1839

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title = "Collocation methods for differential equations with piecewise linear delays",

abstract = "After analyzing the regularity of solutions to delay differential equations (DDEs) with piecewise continuous (linear) non-vanishing delays, we describe collocation schemes using continuous piecewise polynomials for their numerical solution. We show that for carefully designed meshes these collocation solutions exhibit optimal orders of global and local superconvergence analogous to the ones for DDEs with constant delays. Numerical experiments illustrate the theoretical superconvergence results.",

keywords = "Collocation methods, Delay differential equations, Optimal order of superconvergence, Piecewise non-vanishing delays, Regularity of solutions",

author = "Hui Liang and Hermann Brunner",

note = "Part of the work of the first author was carried out while she was a Postdoctoral Research Fellow at Hong Kong Baptist University (FebruaryMay 2009 and March-August 2010); her research was supported by the Hong Kong Research Grants Council (GRF Grant No. HKBU 200207), the National Nature Science Foundation of China (No. 11101130), the Research Fund of the Heilongjiang Provincial Education Department (No. 12511414) and the Heilongjiang University Science Funds for Young Scholar (No. QL201004). ",

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AU - Brunner, Hermann

N1 - Part of the work of the first author was carried out while she was a Postdoctoral Research Fellow at Hong Kong Baptist University (FebruaryMay 2009 and March-August 2010); her research was supported by the Hong Kong Research Grants Council (GRF Grant No. HKBU 200207), the National Nature Science Foundation of China (No. 11101130), the Research Fund of the Heilongjiang Provincial Education Department (No. 12511414) and the Heilongjiang University Science Funds for Young Scholar (No. QL201004).

PY - 2012/9

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N2 - After analyzing the regularity of solutions to delay differential equations (DDEs) with piecewise continuous (linear) non-vanishing delays, we describe collocation schemes using continuous piecewise polynomials for their numerical solution. We show that for carefully designed meshes these collocation solutions exhibit optimal orders of global and local superconvergence analogous to the ones for DDEs with constant delays. Numerical experiments illustrate the theoretical superconvergence results.

AB - After analyzing the regularity of solutions to delay differential equations (DDEs) with piecewise continuous (linear) non-vanishing delays, we describe collocation schemes using continuous piecewise polynomials for their numerical solution. We show that for carefully designed meshes these collocation solutions exhibit optimal orders of global and local superconvergence analogous to the ones for DDEs with constant delays. Numerical experiments illustrate the theoretical superconvergence results.

KW - Collocation methods

KW - Delay differential equations

KW - Optimal order of superconvergence

KW - Piecewise non-vanishing delays

KW - Regularity of solutions

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AN - SCOPUS:84870437818

SN - 1534-0392

VL - 11

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JO - Communications on Pure and Applied Analysis

JF - Communications on Pure and Applied Analysis

IS - 5

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Collocation methods for differential equations with piecewise linear delays

Abstract

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